Page 439 - Contributed Paper Session (CPS) - Volume 2

CPS1917 Trijya S.
of and and consequently, an estimate of the mixing proportion as
̂
̂
2
1
follows:
− ̂
= 1 2
̂
̂
−
2
1
However, in mixture models, the behavior of tails is important, since these
are normally long or thicker tail distributions. By considering only the first
three raw moments, Rider ignored the long tail effect of the mixture model.
To incorporate information about the tail behavior, use of the fourth moment
is essential. In this paper, a methodology is proposed to address these issues.
We propose to obtain the estimates of theoretical moments based on a simple
quadrature formula of numerical integration and the extrapolated tail end
contributions to the moments. We further incorporate the nature of tail
behavior by using the fourth moment in the estimation of the parameters of
equation (1). We have applied the proposed methodology to estimate the
parameters of an extremely useful model used in pharmacokinetic analysis
that is used to study the behavior of drugs in the bodies of human beings and
animals. We have also compared the performance of our methods to that of
an existing method used by Shah (1973), for which the data and estimates are
available.

2. Methodology
Without loss of generality, we assume that in (1),
> . The raw moment of (1) is given by
̂
̂
ℎ
1
2

∞
= ∫ (| , , ). (3)
′

2
1

0
We propose to estimate the integral involved in (3) using a simple trapezoidal
interpolation formula of numerical integration. For this we use an estimate of
density based on a histogram of the data = 1, . . . , . We divide the
,
domain space of data values into bins (class intervals) and count the number
of sample points falling in each bin. Let ℎ () be a smooth function
representing the curve on which top mid points of the bars erected on the
bins lie. The function ℎ () would be close to the kernel function of density
∞
from which the sample has been drawn for all . Let = ∫ ℎ () be the

0
area under curve (AUC). Now, if we know the estimate of AUC, say , then
̂
ℎ ( ) is the estimate of the density at point . The AUC is given by
̂

∞
= ∫ ℎ () = + (4)
1
2
0
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